Problem: What is the remainder when $333^{333}$ is divided by $11$?
Answer: We use the property that $a \equiv b \pmod{m}$ implies $a^c \equiv b^c \pmod{m}$.

$333 \equiv 3 \pmod{11}$, therefore $333^{333} \equiv 3^{333} \pmod{11}$.

Since $3^5 \equiv 1 \pmod{11}$, we get that $333^{333} \equiv 3^{333}=3^{5 \cdot 66 +3}=(3^5)^{66} \cdot 3^3 \equiv 1^{66} \cdot 27 \equiv \boxed{5} \pmod{11}$.